A worked basic example of a survival analysis in R

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Background

I used the rotterdam dataset from the survival package, which includes 2982 primary breast cancers patients whose records were included in the Rotterdam tumor bank.

This dataset records two events: disease relapse and death. Follow-up time is provided for each, with the origin being the time of the primary surgery.

In the article from which this data stems, the authors restricted their dataset to the 1546 patients who had node-positive disease (“Since the validation dataset comprised only node-positive patients and nodal status is an important prognostic factor, we omitted the node-negative patients to create the derivation dataset”). I decided to also restrict the dataset to only node-positive patients.

See:
Royston, P., Altman, D.G. External validation of a Cox prognostic model: principles and methods. BMC Med Res Methodol 13, 33 (2013). https://doi.org/10.1186/1471-2288-13-33

Prognostic factors

Quoted paragraph from the article:
“Candidate prognostic variables in the breast cancer datasets were age at primary surgery (age, years), menopausal status (meno, 0 = premenopausal, 1 = postmenopausal), tumour size (size), tumour grade (grade), number of positive lymph nodes (nodes), progesterone receptors (pgr, fmol/l), oestrogen receptors (er, fmol/l), hormonal treatment (hormon, 0 = no, 1 = yes), and chemotherapy (chemo). Tumour size (mm) was not available as a continuous variable in the Rotterdam dataset, therefore a standard coding was used; the base category was ≤ 20 mm and two dummy variables were used, namely 20 to 50 mm (sized1) and > 50 mm (sized2). We excluded grade, since it was measured according to a different protocol in the two datasets, and chemo, since all patients in the validation dataset received chemotherapy.”

# install.packages("pacman")
pacman::p_load(
        here,
        tidyverse,
        survival,
        survminer,
        ggsurvfit,
        janitor,
        DT,
        flextable,
        patchwork # easily combine plots
)

# Custom package with useful function for summarising dataframe
# pak::pak("UEP-HUG/UEPtools")

# Load in the rotterdam dataset from the survival package
rotterdam <- survival::rotterdam |> 
        # filter out node-negative patients
        filter(nodes > 0) # as "nodal status is an important prognostic factor, we omitted the node-negative patients to create the derivation dataset"

# Add in the variable description
rotterdam_variables <- tribble(
        ~name, ~value,
        "pid", "patient identifier", 
        "year", "year of surgery", 
        "age", "age at surgery", 
        "meno", "menopausal status (0= premenopausal, 1= postmenopausal)", 
        "size", "tumor size, a factor with levels <=20, 20-50, >50", 
        "grade", "differentiation grade", 
        "nodes", "number of positive lymph nodes", 
        "pgr", "progesterone receptors (fmol/l)", 
        "er", "estrogen receptors (fmol/l)", 
        "hormon", "hormonal treatment (0=no, 1=yes)", 
        "chemo", "chemotherapy", 
        "rtime", "days to relapse or last follow-up", 
        "recur", "0= no relapse, 1= relapse", 
        "dtime", "days to death or last follow-up", 
        "death", "0= alive, 1= dead"
)

# Define a function for generating a clean table from the KM estimates
KM_table <- function(x){
        x |> # the survfit2() object
                ggsurvfit::tidy_survfit(times = seq(1000,7000,1000)) |> 
                dplyr::relocate(std.error, .after = conf.low) |> 
                dplyr::mutate(across(c(estimate:conf.low), ~round(.x,2))) |> 
                dplyr::mutate(std.error = round(std.error, 4)) |> 
                dplyr::select(-c(estimate_type:conf.level)) |> 
                flextable::flextable() |> 
                flextable::fontsize(size = 9, part = "all") |> 
                autofit() |> 
                flextable::theme_zebra()
}

# Define a function for generating the KM curves
KM_plot <- function(x, strataname = NULL){
        x |> 
                ggsurvfit()+
                add_censor_mark(size = 1.5, alpha = 0.8) +
                add_confidence_interval() +
                add_risktable(risktable_stats = c("n.risk", "cum.censor", "cum.event")) +
                add_quantile(y_value = 0.5, color = "gray50", linewidth = 0.5) +
                scale_ggsurvfit()+
                add_pvalue()+
                labs(title = "Relapse-free survival after surgery",
                     color = strataname,
                     fill = strataname
                )
}

Inspect Rotterdam data

I went through the package’s documentation for the dataset to also include the descriptions of the variables, and used a custom function to summarise the dataset:

print(paste0("Number of patients analyzed: ", nrow(rotterdam)))

[1] "Number of patients analyzed: 1546"

rotterdam |> 
        UEPtools::metadata_generator_any(variable_names = rotterdam_variables) |> 
        select(-Num_Values) |> 
        DT::datatable(
                filter = "top",
                options = list(
                        pageLength = 26
                ),
                rownames = FALSE, # set to FALSE for cleaner look
                class = 'cell-border stripe hover nowrap compact'
        )

Process dataset for downstream analysis

Create new outcome variable rd, where 0= alive without relapse, 1= relapse or death.
Also create rd_time variable where I calculate days to first of relapse, death or last follow-up

rotterdam_cleaned <- rotterdam |> 
        mutate(
                # Convert several variables to factor
                meno = fct_recode(factor(meno), premenopausal = "0", postmenopausal = "1"),
                grade = factor(grade),
                hormon = fct_recode(factor(hormon), No = "0", Yes = "1"),
                chemo = fct_recode(factor(chemo), No = "0", Yes = "1"),
                
                # Add categorical variable for number of nodes
                nodes_cat = factor(
                        case_when(
                                nodes >0 & nodes <4 ~ "1-3",
                                nodes >= 4 & nodes < 9 ~ "4-9",
                                nodes >=9 ~ "9+"
                        )),
                
                # Add categorical variable for number of estrogen receptors
                er_cat = factor(
                        case_when(
                                er <41 ~ "0-40",
                                er >= 41 & er <301 ~ "41-300",
                                er >=301 ~ "301+"
                        ),levels = c("0-40", "41-300", "301+")),
                
                # Add categorical variable for number of progesterone receptors
                pgr_cat = factor(
                        case_when(
                                pgr <21 ~ "0-20",
                                pgr >= 21 & pgr <201 ~ "21-200",
                                pgr >=201 ~ "201+"
                        ),levels = c("0-20", "21-200", "201+")
                        ),
                
                # `rd` where 0= alive without relapse, 1= relapse or death
                rd = case_when(recur==1|death==1 ~ 1, .default = 0),
                # Calculate days to first of relapse, death or last follow-up
                rd_time = case_when(recur==1 ~ rtime, .default = dtime),
                # Add categorical age variable
                age_cat = factor(case_when( 
                        age < 65  ~ "24-64",
                        age >= 65   ~ "65+"))
        )

# Update the rotterdam_variables object
rotterdam_variables <- bind_rows(
        rotterdam_variables,
        tribble(
                ~name, ~value,
                "rd", "0= alive without relapse, 1= relapse or death",
                "rd_time", "days to relapse/death or last follow-up",
                "age_cat", "age category at surgery")
)

Visualise the distribution of numeric variables

p1 <- rotterdam_cleaned |> ggplot()+geom_histogram(aes(x=age))+labs(x="age at surgery")
p2 <- rotterdam_cleaned |> ggplot()+geom_bar(aes(x=size))+labs(x="tumor size")
p3 <- rotterdam_cleaned |> ggplot()+geom_histogram(aes(x=nodes))+labs(x="number of positive lymph nodes")
p4 <- rotterdam_cleaned |> ggplot()+geom_histogram(aes(x=pgr))+labs(x="progesterone receptors (fmol/l)")
p5 <- rotterdam_cleaned |> ggplot()+geom_histogram(aes(x=er))+labs(x="estrogen receptors (fmol/l)")
p6 <- rotterdam_cleaned |> ggplot()+geom_bar(aes(x=hormon))+labs(x="hormonal treatment")
p7 <- rotterdam_cleaned |> ggplot()+geom_bar(aes(x=meno))+labs(x="menopausal status")
p8 <- rotterdam_cleaned |> ggplot()+geom_histogram(aes(x=rd_time))+labs(x="days to relapse/death or last follow-up")

p1+p2+p3+p4+p5+p6+p7+p8+ plot_layout(ncol = 2)

Analysis

Kaplan-Meier curve

Relapse-free survival probability at different timepoints:

# Fit a Surv-type object for right-censored data
rotterdam_surv_fit_rd <-  
        # survival::survfit(Surv(rd_time, rd) ~ 1, data = rotterdam_cleaned)
        # Using survfit2 from the ggsurvfit package allows easier control for 
        # downstream analysis using the ggsurvfit() function:
        ggsurvfit::survfit2(Surv(rd_time, rd) ~ 1, data = rotterdam_cleaned)
# Print its summary at specific times
rotterdam_surv_fit_rd |> KM_table()

time	n.risk	n.event	n.censor	cum.event	cum.censor	estimate	conf.high	conf.low	std.error
1,000	928	608	10	608	10	0.61	0.63	0.58	0.0205
2,000	573	289	66	897	76	0.41	0.44	0.39	0.0307
3,000	321	105	147	1,002	223	0.33	0.35	0.30	0.0384
4,000	112	62	147	1,064	370	0.24	0.27	0.22	0.0547
5,000	24	14	74	1,078	444	0.19	0.22	0.15	0.0970
6,000	3	2	19	1,080	463	0.14	0.22	0.09	0.2196
7,000	1	0	2	1,080	465	0.14	0.22	0.09	0.2196

Restricted Mean Survival Time (RMST): Restricted Mean Survival Time (RMST) is a statistical measure used in survival analysis that offers a clinically meaningful way to compare survival outcomes between groups. It represents the average event-free survival time up to a pre-specified time point, and is defined as the area under the survival curve from time zero up to a specific, pre-defined time point. (using the survival package, I can only print the RMST for single groups or strata, but for more detailed comparisons between groups (e.g. getting the difference in RMST with confidence intervals) I could try more specialised packages, e.g. survRM2.

“The RMST may provide valuable information for comparing two survival curves when the proportional hazards assumption is not met, such as in cases of crossing or delayed separation of survival curves.”
- Han K, Jung I. Restricted Mean Survival Time for Survival Analysis: A Quick Guide for Clinical Researchers. Korean J Radiol. 2022 May;23(5):495-499. doi: 10.3348/kjr.2022.0061

For this analysis, I’ve set the restriction time at 5 years (1,825 days):

print(rotterdam_surv_fit_rd, rmean = 365*5,  print.rmean = TRUE)

Call: survfit(formula = Surv(rd_time, rd) ~ 1, data = rotterdam_cleaned)

        n events rmean* se(rmean) median 0.95LCL 0.95UCL
[1,] 1546   1080   1225      16.1   1441    1301    1583
    * restricted mean with upper limit =  1825 

Overall plot

rotterdam_surv_fit_rd |> 
        ggsurvfit(color = "tomato")+
        add_censor_mark(color = "tomato", size = 1.5, alpha = 0.8) +
        add_confidence_interval(fill = "tomato") +
        add_risktable(risktable_stats = c("n.risk", "cum.censor", "cum.event")) +
        add_quantile(y_value = 0.5, color = "gray60", linewidth = 0.5) +
        scale_ggsurvfit()+
        labs(title = "Relapse-free survival after surgery")

Group comparison: menopausal status (example)

Survival probabilities:

rotterdam_surv_fit_meno <-  ggsurvfit::survfit2(Surv(rd_time, rd) ~ meno, data = rotterdam_cleaned)
rotterdam_surv_fit_meno |> KM_table() |> hline(7)

time	n.risk	n.event	n.censor	cum.event	cum.censor	estimate	conf.high	conf.low	std.error	strata
1,000	524	386	8	386	8	0.58	0.61	0.55	0.0282	postmenopausal
2,000	305	181	38	567	46	0.37	0.41	0.34	0.0433	postmenopausal
3,000	162	68	75	635	121	0.28	0.31	0.25	0.0561	postmenopausal
4,000	51	37	74	672	195	0.19	0.23	0.16	0.0848	postmenopausal
5,000	7	11	33	683	228	0.12	0.17	0.08	0.1896	postmenopausal
6,000	1	1	5	684	233	0.06	0.25	0.01	0.7321	postmenopausal
7,000	0	0	1	684	234					postmenopausal
1,000	404	222	2	222	2	0.65	0.68	0.61	0.0296	premenopausal
2,000	268	108	28	330	30	0.47	0.51	0.43	0.0428	premenopausal
3,000	159	37	72	367	102	0.40	0.44	0.36	0.0515	premenopausal
4,000	61	25	73	392	175	0.32	0.36	0.28	0.0692	premenopausal
5,000	17	3	41	395	216	0.28	0.34	0.23	0.1002	premenopausal
6,000	2	1	14	396	230	0.26	0.33	0.20	0.1327	premenopausal
7,000	1	0	1	396	231	0.26	0.33	0.20	0.1327	premenopausal

Restricted Mean Survival Time: Over 5 years (1825 days)

# Print mean survival time
print(rotterdam_surv_fit_meno, rmean = 365*5, print.rmean = TRUE)

Call: survfit(formula = Surv(rd_time, rd) ~ meno, data = rotterdam_cleaned)

                      n events rmean* se(rmean) median 0.95LCL 0.95UCL
meno=premenopausal  628    396   1298      24.1   1749    1510    2141
meno=postmenopausal 918    684   1174      21.4   1275    1172    1442
    * restricted mean with upper limit =  1825 

Over 10 years (3650 days)

# Print mean survival time
print(rotterdam_surv_fit_meno, rmean = 365*10, print.rmean = TRUE)

Call: survfit(formula = Surv(rd_time, rd) ~ meno, data = rotterdam_cleaned)

                      n events rmean* se(rmean) median 0.95LCL 0.95UCL
meno=premenopausal  628    396   2046      55.2   1749    1510    2141
meno=postmenopausal 918    684   1733      44.4   1275    1172    1442
    * restricted mean with upper limit =  3650 

KM curve:

rotterdam_surv_fit_meno |> KM_plot()

Log-rank test

Use a log-rank test, which accounts for the whole follow-up period. By comparing the observed number of events in each group with the number that would be expected if event rates were the same, a chi-squared test is used to determine if any observed differences are statistically meaningful.

survdiff(Surv(rd_time, rd) ~ meno, rho = 0, data = rotterdam_cleaned)

Call:
survdiff(formula = Surv(rd_time, rd) ~ meno, data = rotterdam_cleaned, 
    rho = 0)

                      N Observed Expected (O-E)^2/E (O-E)^2/V
meno=premenopausal  628      396      479      14.4        26
meno=postmenopausal 918      684      601      11.5        26

 Chisq= 26  on 1 degrees of freedom, p= 3e-07 

Regression modelling (CoxPH)

Visually inspect individual KM curves for covariates

Numeric covariates are categorized here only to aid visualisation

rotterdam_surv_fit_age_cat <-  survfit2(Surv(rd_time, rd) ~ age_cat, data = rotterdam_cleaned)
rotterdam_surv_fit_age_cat |> KM_plot(strataname = "Age category")

rotterdam_surv_fit_size <-  survfit2(Surv(rd_time, rd) ~ size, data = rotterdam_cleaned)
rotterdam_surv_fit_size |> KM_plot(strataname = "Tumor size")

rotterdam_surv_fit_hormon <-  survfit2(Surv(rd_time, rd) ~ hormon, data = rotterdam_cleaned)
rotterdam_surv_fit_hormon |> KM_plot(strataname = "Hormonal treatment")

rotterdam_surv_fit_nodes_cat <-  survfit2(Surv(rd_time, rd) ~ nodes_cat, data = rotterdam_cleaned)
rotterdam_surv_fit_nodes_cat |> KM_plot(strataname = "Number of nodes")

rotterdam_surv_fit_pgr_cat <-  survfit2(Surv(rd_time, rd) ~ pgr_cat, data = rotterdam_cleaned)
rotterdam_surv_fit_pgr_cat |> KM_plot(strataname = "progesterone receptors (fmol/l)")

rotterdam_surv_fit_er_cat <-  survfit2(Surv(rd_time, rd) ~ er_cat, data = rotterdam_cleaned)
rotterdam_surv_fit_er_cat |> KM_plot(strataname = "estrogen receptors (fmol/l)")

Cox model specification

This is the full model using the variables specified in the original article (see section “Prognostic factors” above)

rotterdam_cox <- 
        coxph(Surv(rd_time, rd) ~ age + meno + size + nodes + pgr + er + hormon,
              data = rotterdam_cleaned
        )
summary(rotterdam_cox)

Call:
coxph(formula = Surv(rd_time, rd) ~ age + meno + size + nodes + 
    pgr + er + hormon, data = rotterdam_cleaned)

  n= 1546, number of events= 1080 

                         coef  exp(coef)   se(coef)      z Pr(>|z|)    
age                 0.0045330  1.0045433  0.0039914  1.136  0.25608    
menopostmenopausal  0.2418071  1.2735485  0.1065947  2.268  0.02330 *  
size20-50           0.2855600  1.3305069  0.0738251  3.868  0.00011 ***
size>50             0.6124397  1.8449269  0.0941898  6.502 7.92e-11 ***
nodes               0.0558619  1.0574516  0.0052392 10.662  < 2e-16 ***
pgr                -0.0002002  0.9997998  0.0001237 -1.618  0.10560    
er                 -0.0002603  0.9997398  0.0001250 -2.082  0.03733 *  
hormonYes          -0.3228457  0.7240856  0.0809431 -3.989 6.65e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

                   exp(coef) exp(-coef) lower .95 upper .95
age                   1.0045     0.9955    0.9967    1.0124
menopostmenopausal    1.2735     0.7852    1.0334    1.5695
size20-50             1.3305     0.7516    1.1513    1.5376
size>50               1.8449     0.5420    1.5339    2.2190
nodes                 1.0575     0.9457    1.0466    1.0684
pgr                   0.9998     1.0002    0.9996    1.0000
er                    0.9997     1.0003    0.9995    1.0000
hormonYes             0.7241     1.3811    0.6179    0.8486

Concordance= 0.66  (se = 0.008 )
Likelihood ratio test= 235.5  on 8 df,   p=<2e-16
Wald test            = 266.1  on 8 df,   p=<2e-16
Score (logrank) test = 277.2  on 8 df,   p=<2e-16

Forest plot

survminer::ggforest(rotterdam_cox, data = rotterdam_cleaned)

Assumptions: Proportional hazards

Schoenfeld residuals test:

• For each covariate and globally, it tests the null hypothesis that the PH assumption holds (i.e., the slope of Schoenfeld residuals vs. time is zero).
  
• A small p-value (e.g. < 0.05) for a covariate suggests that the PH assumption is violated for that variable.
  
• The ‘GLOBAL’ test indicates if the assumption is violated for the overall model.

rotterdam_test <- cox.zph(rotterdam_cox)
rotterdam_test

          chisq df       p
age     4.05333  1 0.04408
meno    0.00248  1 0.96026
size   13.17300  2 0.00138
nodes   6.73579  1 0.00945
pgr    15.02219  1 0.00011
er     24.55857  1 7.2e-07
hormon  2.99857  1 0.08334
GLOBAL 61.45472  8 2.4e-10

The ‘GLOBAL’ test appears to show that the PH assumption is violated for the overall model, and also appears to be violated for the age, size, nodes, and er variables.

Visual inspection of the test:

• Plots the scaled Schoenfeld residuals against transformed time for each covariate.
  
• A non-horizontal line with a non-zero slope suggests a violation of the PH assumption.

wrap_plots(survminer::ggcoxzph(rotterdam_test), ncol =2)

While the results of the test of the proportional hazards assumption imply violation of the proportional hazards assumption, after visually inspecting the Schoenfeld residuals the validity of the assumption may still be argued to hold. Still, next steps may involve some combination of transformation of variables and/or consideration of including time-dependent interaction terms.

We could additionally further explore comparing estimates of Restricted Mean Survival Time (RMST), e.g. see:
Han K, Jung I. Restricted Mean Survival Time for Survival Analysis: A Quick Guide for Clinical Researchers. Korean J Radiol. 2022 May;23(5):495-499. doi: 10.3348/kjr.2022.0061

“We conclude that the hazard ratio cannot be recommended as a general measure of the treatment effect in a randomized controlled trial, nor is it always appropriate when designing a trial. Restricted mean survival time may provide a practical way forward and deserves greater attention.”
- Royston, P., Parmar, M.K. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Med Res Methodol 13, 152 (2013). https://doi.org/10.1186/1471-2288-13-152